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Hydrogen-exact exchange factor binds Rydberg states in semilocal DFT

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2026-07-09 03:43 UTC pith:YNY7N7TE

load-bearing objection First semilocal exchange functional to recover the -1/r asymptotic tail and bound Rydberg states in s-shell atoms, via a complete solution of the 1993 Gill-Pople equation. the 1 major comments →

arxiv 2607.07648 v1 pith:YNY7N7TE submitted 2026-07-08 physics.comp-ph cond-mat.mtrl-sci

Semilocal exchange functionals from the exact-exchange condition for the hydrogen atom: Hydrogenic exactness and recovery of Rydberg-like bound states

Fumihiro Imoto This is my paper
classification physics.comp-ph cond-mat.mtrl-sci PACS 31.15.E-71.15.Mb
keywords exchangefactorfunctionalgp93atomboundfunctionalshydrogen
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

Semilocal density functionals such as PBE and SCAN produce exchange potentials that decay exponentially outside atoms and molecules, lacking the correct Coulombic tail of -1/r. This structural defect means no existing semilocal functional supports bound Rydberg-like virtual states in atoms like helium, where PBE through SCAN bind zero. The paper revisits a 1993 ordinary differential equation derived by Gill and Pople, which specifies the condition a gradient-corrected exchange functional must satisfy to reproduce the exact exchange potential on the hydrogen atom. The author solves this equation completely as an inverse problem, obtaining a unique enhancement factor called the GP93 factor. This factor is exact on the hydrogen 1s density by construction and, through uniform coordinate scaling, yields the exact eigenvalue -Z^2/2 for every hydrogenic 1s ion. Its large-gradient growth follows the form O[s(ln s)^{2/3}], which is the hydrogen-exact structure and is distinct from the s*ln(s) form used in prior attempts at improved asymptotics. The GP93 factor contains no empirical parameters. To prevent the unbounded enhancement factor from destabilizing many-electron systems, the author embeds it in a switched meta-GGA form. A kinetic-energy-density indicator alpha detects one-electron-like regions (where alpha equals zero) and activates the GP93 term only there, while reverting to PBE in many-electron regions. A reduced-Laplacian indicator q further distinguishes atomic tails (where q is large and positive) from covalent bond centers (where q is small or negative), suppressing the divergent branch at bond centers without disturbing the atomic tail. The resulting functional produces Rydberg-like series of bound virtual Kohn-Sham states in all five tested systems whose outermost shell is a one-electron-like s shell: H, He, and Li, Na, K with large-core pseudopotentials. For p-shell atoms Ne and Ar, the switch closes by design and no bound states appear. The HOMO eigenvalue moves toward the experimental ionization energy with a mean absolute error of 0.79 eV across the five s-shell systems, compared to 3.3-4.2 eV for PBE, SCAN, and B3LYP. The author also shows that the orbital-dependent kinetic-energy-density indicator can be replaced by an orbital-free kinetic-energy-density functional with a von Weizsacker lower bound, rendering the entire switch a pure density and

Core claim

The central object is the GP93 factor, defined by solving the Gill-Pople equation as an inverse problem. It is an enhancement factor for semilocal exchange that is exact on the hydrogen 1s density by construction, exact on all hydrogenic 1s ions by coordinate scaling, and has the hydrogen-exact large-gradient growth O[s(ln s)^{2/3}]. When embedded in a switched functional that activates the factor only in one-electron-like regions detected by the alpha indicator and further refined by the reduced Laplacian q, the functional produces bound Rydberg-like virtual Kohn-Sham states in all five tested s-shell systems (H, He, Li, Na, K) where PBE through SCAN bind none in He. The domain of applicabl

What carries the argument

GP93 factor (solution of the Gill-Pople equation as inverse problem); alpha indicator (kinetic-energy-density ratio detecting one-electron character); q indicator (reduced Laplacian distinguishing atomic tails from covalent bond centers); triple-switched enhancement factor combining PBE with the GP93 factor; orbital-free alpha_PC replacing the orbital-dependent kinetic-energy density with a Perdew-Constantin-type functional

Load-bearing premise

The alpha-switch and q-switch, with a few calibrated parameters, are assumed to cleanly separate one-electron-like atomic tails from many-electron regions and covalent bond centers. This separation is demonstrated only non-self-consistently on fixed densities for molecules, and the self-consistent q-switch is unstable in Gaussian bases. The entire molecular and solid-state extension rests on the unverified premise that a real-space self-consistent implementation will converge

What would settle it

A self-consistent real-space implementation of the triple-switched functional that fails to converge, or that converges but does not reproduce the bound Rydberg states and HOMO-ionization-energy improvements shown in the fixed-density and Gaussian-basis calculations, would falsify the central claim that the GP93 factor can be embedded in a stable, practical semilocal functional for general systems.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • A self-consistent real-space pseudopotential implementation of the triple-switched functional could recover Rydberg-like bound states and improved ionization energies at semilocal computational cost, without nonlocal exact exchange.
  • The alpha-switch design principle predicts that only systems with one-electron-like s-shell HOMOs benefit from the asymptotic correction, while p-shell atoms and many-electron regions revert to PBE, fixing the domain of applicability by construction.
  • Large-core pseudopotentials play a constructive role by rendering alkali valence electrons one-electron-like, suggesting a natural affinity between this functional and real-space pseudopotential codes.
  • The orbital-free replacement of the kinetic-energy-density indicator removes the need for a generalized Kohn-Sham solver, making the entire switch a pure density functional with a standard local exchange potential.
  • The -1/r correction is switched off in smooth bulk interiors and acts in low-density one-electron-like regions such as the vacuum side of surfaces, potentially affecting adsorption energetics where image-potential-like asymptotes matter.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

1 major / 6 minor

Summary. This manuscript solves the Gill-Pople (GP93) ordinary differential equation for a GGA exchange functional that reproduces the exact exchange potential of the hydrogen atom, obtaining the enhancement factor $w(z)$ in closed form via variation of constants (Eq. 7) with Frobenius matching at $z=0$ (Eq. 10). The resulting GP93 factor is parameter-free, exhibits the hydrogen-exact large-gradient growth $O[s(¥ln s)^{2/3}]$ (Proposition 3), and is exact on hydrogenic 1s ions by coordinate scaling (Theorem 2). The factor is embedded in a switched meta-GGA form (Eq. 16) using the indicator $¥alpha=(¥tau-¥tau_W)/¥tau_{¥rm unif}$ to confine the divergent branch to one-electron-like regions. The central empirical result is that this functional produces Rydberg-like series of bound virtual Kohn-Sham states in all five tested s-shell systems (H, He, Li, Na, K) where PBE through SCAN bind none in He, while p-shell atoms (Ne, Ar) are correctly left unmodified. A triple switch adding the reduced Laplacian $q$ (Eq. 18) is introduced to suppress divergence at covalent bond centers, and an orbital-free replacement for $¥alpha$ via a Perdew-Constantin-type kinetic-energy-density functional (Eq. 28) is proposed to eliminate the need for a generalized Kohn-Sham solver.

Significance. The paper ships a genuinely parameter-free, analytically derived exchange ingredient (the GP93 factor) with a rigorous hydrogen-exactness theorem (Theorem 1) and a scale-covariance eigenvalue theorem (Theorem 2) verified numerically on He+ and Li2+ not used in calibration (Sec. VIII: 54.44 vs 54.42 eV). The asymptotic growth $O[s(¥ln s)^{2/3}]$ (Proposition 3, Eq. 12) is a concrete, falsifiable prediction distinct from the AK13 $s¥ln s$ form. The recovery of 4-8 bound Rydberg-like states across five s-shell systems (Tables II-III), with zero bound states in p-shell atoms by design, is a clear qualitative advance over existing semilocal functionals. The Frobenius matching is stable to eight digits (Appendix B), and implementation verification includes derivative checks to $10^{-9}$ and energy-minimum confirmation (Appendix C). The scope is honestly delimited: no thermochemistry claims, no p-shell HOMO improvement, no response properties.

major comments (1)
  1. §VI.B, Table II, and the He Rydberg result: The headline result of 8 bound Rydberg states in He is obtained with the alpha-switched form of Eq. (16), not the triple-switched form of Eq. (18). The q-switch is introduced (Sec. IV.A) because alpha alone is insufficient to suppress the divergent F_mix branch in low-density, large-s, alpha≈0 regions such as covalent bond centers. The manuscript argues (Eq. 21) that the q-switch should not perturb the He atomic tail because q→+∞ there, making D(q)→1. However, this argument is verified only non-self-consistently on fixed densities (Sec. VI.D). A self-consistent q-switched calculation on He—confirming that the bound-state count and HOMO are preserved when the full triple switch is active—would close the logical gap between the functional form advocated for molecules/solids (Eq. 18) and the functional form used to produce the central atomic Ryd伯格
minor comments (6)
  1. §V, Computational Methods: The basis-set dependence of the Rydberg state count is acknowledged but not quantified. A convergence study showing how the count of bound states in He changes from aug-cc-pVDZ through aug-cc-pV5Z would strengthen the claim that the qualitative contrast (zero vs. eight) is basis-robust.
  2. Table I: Only H, He+, and Li2+ are listed. The text mentions Z=1-5 in Sec. VIII, but Table I shows only three systems. Adding Z=4,5 would complete the demonstration of Theorem 2.
  3. §VII, Eq. (28): The large-p damping factor with p_c=5 is acknowledged as 'not a parameter-free consequence of the GP93 construction.' This is an additional empirical parameter that should be listed alongside the switching parameters in the parameter inventory for full transparency.
  4. Fig. 1 caption: The y-axis label 'Mean absolute error' should specify units (eV) for consistency with Eq. (14).
  5. §III, Eq. (13): The outer switch parameters (p, s_0) are described as 'calibrated' rather than 'empirical.' While the distinction is explained, the phrasing may invite confusion; a brief note that 'calibrated' means 'determined by the functional's design purpose' would help.
  6. Reference [3] (Gill and Pople 1993) is central to the entire construction but is cited only in the introduction and Sec. II. A brief note on how the present solution relates to GP93's original (incomplete) treatment would provide useful historical context.

Simulated Author's Rebuttal

1 responses · 1 unresolved

The referee identifies one major issue: the headline He Rydberg result (8 bound states) is obtained with the alpha-switched form (Eq. 16), while the triple-switched form (Eq. 18) including the q-switch is advocated for molecules/solids but verified only non-self-consistently. The referee requests a self-consistent q-switched calculation on He to confirm the bound-state count and HOMO are preserved under the full triple switch. We agree this is a logical gap and will address it by performing the requested calculation or, where self-consistent implementation in Gaussian bases is not feasible, by providing a detailed non-self-consistent verification on the self-consistent alpha-switched density and revising the manuscript to state the limitation explicitly.

read point-by-point responses

  1. Referee: §VI.B, Table II, and the He Rydberg result: The headline result of 8 bound Rydberg states in He is obtained with the alpha-switched form of Eq. (16), not the triple-switched form of Eq. (18). The q-switch is introduced (Sec. IV.A) because alpha alone is insufficient to suppress the divergent F_mix branch in low-density, large-s, alpha≈0 regions such as covalent bond centers. The manuscript argues (Eq. 21) that the q-switch should not perturb the He atomic tail because q→+∞ there, making D(q)→1. However, this argument is verified only non-self-consistently on fixed densities (Sec. VI.D). A self-consistent q-switched calculation on He—confirming that the bound-state count and HOMO are preserved when the full triple switch is active—would close the logical gap between the functional form advocated for molecules/solids (Eq. 18) and the functional form used to produce the central atomic Ryd伯格

    Authors: The referee is correct that the headline He Rydberg result (Table II, 8 bound states) is obtained with the alpha-switched form of Eq. (16), and that the q-switch of Eq. (18) is verified only non-self-consistently. We agree that a self-consistent q-switched calculation on He would close the logical gap between the functional form advocated for general use and the one used to produce the central atomic result. We will address this in revision. The key question is whether the q-switch, when active self-consistently, preserves the He bound-state count and HOMO. Our analytical argument (Eq. 21) shows that in the He atomic tail q→+∞, so D(q)→1 and the q-switch is saturated; the derivative ∂D/∂q_c vanishes there, meaning the q-switch is inert in the region that determines the Rydberg levels. The referee's concern is whether this holds self-consistently, not just on fixed densities. We will attempt the self-consistent q-switched calculation on He. However, we note that the v_lapl term associated with q is numerically unstable in Gaussian bases (as discussed in Sec. IV.A and Sec. VIII), which is why the q-switch was demonstrated non-self-consistently in the original submission. If a stable self-consistent Gaussian implementation proves infeasible, we will: (1) provide a non-self-consistent evaluation of the full triple-switched functional on the self-consistent alpha-switched He density, showing that the exchange potential tail and bound-state spectrum are unchanged when D(q) is applied; (2) revise the manuscript to state explicitly that the self-consistent q-switched verification on He is a necessary next step requiring real-space implementation, and that the current evidence supports—but does not definitively confirm—that the triple switch preserves the atomic Rydberg result. revision: partial

standing simulated objections not resolved
  • A fully self-consistent q-switched calculation on He may not be achievable within the Gaussian-basis framework used in this work, because the v_lapl term (functional derivative of the exchange energy with respect to ∇²n) is numerically unstable in Gaussian bases. This is the same difficulty faced by Laplacian-level meta-GGAs and is the reason the q-switch was demonstrated non-self-consistently in the original submission. A definitive self-consistent verification likely requires a real-space grid implementation, which is beyond the scope of the current manuscript. If the self-consistent calculation cannot be stabilized, the logical gap identified by the referee can be narrowed but not fully closed in this revision.

Circularity Check

2 steps flagged

GP93 factor is parameter-free and independently verified; switching parameters are fitted to the target MAE and then the same MAE is reported as a result, but the central Rydberg-binding claim has independent content.

specific steps
  1. fitted input called prediction [Sec. III, Eqs. (13)-(14), Fig. 1]
    "The parameters (p, s 0) of the outer switch are not an empirical fit: they are calibrated against the very purpose of the functional—the recovery of bound Rydberg states and the agreement of −ε HOMO with experimental ionization energies. We minimize the mean absolute error MAE = 1/5 Σ |−εHOMO − I exp| over the five systems with one-electron-like s-shell HOMOs (H, He, Li, Na, K)... The optimum over the stable domain is (p⋆, s⋆0) = (8, 0.16), MAE = 0.80 eV, which we use throughout."

    The outer-switch parameters (p=8, s0=0.16) are obtained by minimizing the MAE of −εHOMO versus experimental IP over exactly the five s-shell systems (H, He, Li, Na, K). The paper then reports this same MAE = 0.80 eV (Table III, Fig. 4) as evidence of the functional's quality: 'The mean absolute HOMO–IP errors are PBE 4.19, SCAN 3.95, B3LYP 3.30, and F_mix 0.79 eV, four to five times better than the other functionals.' The 0.79 eV is the fitted minimum by construction; it cannot be worse than the competitors on this set because the parameters were chosen to minimize it. The paper's claim that this is 'not an empirical fit' is contradicted by the procedure itself. However, the qualitative Rydberg-binding result (8 bound states in He where PBE/SCAN bind none) is not directly the fitted target

  2. self definitional [Sec. IV, Eqs. (16)-(20), and Sec. VI.B/VI.D]
    "A prediction of the domain of applicability follows immediately: the asymptotic improvement reaches the HOMO only if the HOMO itself is one-electron-like (α≈0), i.e., only in systems whose outermost shell is occupied by a single ns electron. For p-shell HOMOs (α > 0) the switch suppresses the GP93 term and no effect appears. This prediction is verified numerically in Sec. VI."

    The switch G(α) = exp[−(α/wα)²] is defined so that G(0)=1 (GP93 term fully active) and G(α>0)→0 (GP93 term suppressed). The 'prediction' that s-shell systems (α≈0) show Rydberg states while p-shell systems (α>0) do not is then verified on Ne and Ar. But this outcome is forced by the switch definition: the functional is constructed to activate only where α≈0, so observing that it activates only where α≈0 is tautological. The paper acknowledges this: 'the domain of applicability follows from the design itself.' This is a minor circularity because the α≈0 condition for one-electron regions is an exact physical property (τ=τW for single orbitals), not an arbitrary ansatz, so the switch design has independent physical motivation.

full rationale

The GP93 factor itself—the paper's central nonempirical ingredient—is derived from the Gill-Pople equation (an independent mathematical constraint from 1993, cited as Ref. [3] with no author overlap) and contains no fitted parameters. Theorem 1 (hydrogen-exactness) holds by construction of solving that equation, which the paper openly states. Theorem 2 (hydrogenic eigenvalue exactness) follows from coordinate scaling and is independently verified on He+ and Li2+ (54.44 vs 54.42 eV, 122.50 vs 122.45 eV)—systems not used in calibration. The main empirical result (Rydberg-like bound states in He where PBE/SCAN bind none) is a qualitative outcome not directly equivalent to the fitted MAE. The circularity is limited to: (1) the s0 calibration minimizing MAE on the same five systems whose MAE is then reported as a quality metric, and (2) the s-shell/p-shell selectivity being a consequence of the switch definition rather than an independent prediction. Neither reduces the central claim to its inputs. The paper is notably transparent about both points, explicitly calling the domain restriction 'fixed by design' and the MAE a 'calibration.' This is a borderline-fitted but not fundamentally circular paper.

Axiom & Free-Parameter Ledger

8 free parameters · 5 axioms · 0 invented entities

The paper introduces no new physical entities, particles, forces, or dimensions. The GP93 factor w(z) is a mathematical function derived from an ODE, not a postulated entity. The switching functions (G, D, chi) are constructed from established meta-GGA indicators (alpha, q, s). The orbital-free alpha_PC uses the existing Perdew-Constantin KEDF [15]. All free parameters are either calibrated switching parameters (s0, p, w_alpha, q_c, w_q, p_c) or determined by internal mathematical consistency (c0). The PC07 parameters (a, b) are inherited from prior literature.

free parameters (8)
  • c0 = w(0) = 0.68928799
    Determined by matching the z>0 representation (Eq. 7) to the Frobenius series at z=0. Not fitted to data but determined by internal consistency of the GP93 solution. Stable to eight digits over matching points z_m in {2,3,4,5}.
  • p (outer switch exponent) = 8
    Calibrated; results saturate for p>=8 (Fig. 1). Chosen to protect GE2 and ensure doubly-exponential suppression of 1-chi on hydrogen density.
  • s0 (outer switch scale) = 0.16
    Calibrated by minimizing MAE of -epsilon_HOMO vs experimental IP over H, He, Li, Na, K (Fig. 1, Eq. 14). SCF diverges for s0 < 0.15.
  • w_alpha (alpha switch width) = 0.10
    Controls the Gaussian G(alpha)=exp[-(alpha/w_alpha)^2]. Not explicitly calibrated against data; chosen so G'(0)=0 does not perturb one-electron regions.
  • q_c (q-switch center) = 1
    Calibrated; separates atomic tails (q>>q_c) from covalent bond centers (q<q_c). Tested on fixed densities for ten molecules (Table V).
  • w_q (q-switch width) = 0.6
    Controls the tanh width in D(q). Not explicitly calibrated; chosen for smooth switching.
  • p_c (orbital-free alpha damping) = 5
    Ad hoc damping in Eq. 28 forcing alpha_PC -> 0 as p -> infinity. Author explicitly states this is 'not a parameter-free consequence of the GP93 construction.'
  • a, b (PC07 interpolation) = a=0.5389, b=3
    Parameters of the Perdew-Constantin KEDF interpolation function f_ab (Eq. 27). Inherited from the PC07 reference [15], not introduced by this paper.
axioms (5)
  • domain assumption The Kohn-Sham exchange potential should satisfy v_x(r) -> -1/r as r -> infinity for finite systems.
    Invoked in Sec. I (Eq. 1) as the exact asymptotic condition. Standard DFT result, not specific to this paper.
  • standard math The GGA exchange energy takes the form E_x = A_x * integral n^{4/3} F(s) with s = |grad n| / [2(3*pi^2)^{1/3} n^{4/3}].
    Standard GGA form (Eq. 2). Invoked throughout as the semilocal energy ansatz.
  • domain assumption The meta-GGA indicator alpha = (tau - tau_W) / tau_unif exactly detects one-electron regions (alpha=0 iff single-orbital).
    Invoked in Sec. IV (Eq. 15). Standard meta-GGA result from SCAN/TPSS literature. The paper relies on alpha=0 being a sufficient detector of one-electron character for the switch to work.
  • standard math Uniform coordinate scaling n_kappa(r) = kappa^3 n(kappa r) preserves the form of scale-invariant functionals: E_x[n_kappa] = kappa * E_x[n].
    Invoked in Theorem 2 (Sec. II). Standard DFT scaling result used to extend hydrogen exactness to hydrogenic ions.
  • domain assumption LYP correlation vanishes for one-electron densities, preserving hydrogenic exactness.
    Invoked in Proposition 6 (Sec. IV). Property of the LYP functional [6]. Used to justify LYP as the recommended correlation companion.

pith-pipeline@v1.1.0-glm · 21169 in / 4239 out tokens · 864249 ms · 2026-07-09T03:43:02.904501+00:00 · methodology

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read the original abstract

In semilocal density functionals the exchange potential decays too rapidly outside atoms and molecules and lacks the correct $-1/r$ tail; as a consequence, functionals from PBE to modern meta-GGAs such as SCAN support no bound Rydberg-like state of an atom. We revisit the gradient-corrected exchange functional that Gill and Pople (GP93) constructed to reproduce the exact exchange potential of the hydrogen atom, solve their equation as an inverse problem, and use the resulting enhancement factor -- the GP93 factor -- as a tail-generating ingredient of a switched semilocal functional. The GP93 factor is exact on the hydrogen $1s$ density and, by uniform coordinate scaling, on hydrogenic $1s$ ions, where it yields the eigenvalue $-Z^2/2$ exactly (hydrogenic exactness). Its large-gradient growth has the hydrogen-exact form $O[s(\ln s)^{2/3}]$, distinct from the $s\ln s$ form of Armiento and K\"ummel. The factor is combined with a kinetic-energy-density indicator so that its divergent branch acts only in one-electron-like regions; the GP93 ingredient is nonempirical, while a few switching parameters are calibrated to balance tail recovery against self-consistent-field stability. The resulting functional produces Rydberg-like series of bound virtual Kohn--Sham states in systems whose outermost shell is a one-electron-like $s$ shell -- all-electron H and He, and Li, Na, and K with large-core pseudopotentials -- and moves the highest-occupied eigenvalue toward the experimental ionization energy. For $p$-shell atoms (Ne, Ar) the switch closes, so the domain of applicability is fixed by design. Fixed-density tests indicate that a reduced-Laplacian switch distinguishes atomic tails from covalent bond centers, and that a density-only kinetic-energy functional renders the entire switch orbital-independent, so that no generalized Kohn--Sham solver is required.

Figures

Figures reproduced from arXiv: 2607.07648 by Fumihiro Imoto.

Figure 1
Figure 1. Figure 1: FIG. 1. Mean absolute error of the Rydberg series versus the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. (a) The GP93 auxiliary function [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Helium atom. (A) [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Atomic systematics, all-functional comparison (number above each bar = bound Rydberg orbitals). [PITH_FULL_IMAGE:figures/full_fig_p008_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Binding-region potential curve of H [PITH_FULL_IMAGE:figures/full_fig_p009_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Radial exchange potential [PITH_FULL_IMAGE:figures/full_fig_p010_6.png] view at source ↗

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Reference graph

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